“那位画师是什么人？”景容开口。 “她叫纪云舒，是咱们锦江城出了名的画师。”县太爷答。 “如何出名？”
Does it mean
How famous is she?
How did she become famous?
What is she famous for?
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I believe the sentence is slightly ambiguous and both the first and third interpretations you proposed are possible. The first ("How famous is she") asks for the degree of fame, and the third ("How is she famous") asks for the manner of fame. It would be natural if the answer to the question were, say, a description of how the painter's work is sought after across the whole province and how her fame trumps that of other well-known painters, or a description of the painter's exceptional skill.
In fact, the author intends the third interpretation, because they continued to recount that the painter was able to reconstruct the faces of deceased persons accurately,
The reason why the second interpretation is impossible is a bit subtle. I am tempted to say that the plain question 如何出名 can only be asking about a factual or habitual statement. To indicate past, one needs to "topicalize" an action with a finite temporal extent, so (at least) the particle 的 should be added, turning it into 如何出名的.
出了名的画师 = famous painter
如何出名？” = how famous?
It means: "how famous is she?" (to what degree she is famous, city-wise, nationally, or worldwide?)
Since they knew she is a famous painter already, there was no reason for asking "what she is famous for" ( 她凭什么出名的?/ 她出名在什么地方?)
If he's asking "How did she becomes famous" the question would be 她是怎樣出名的?/ 她是如何出名的? (how did she become famous)
出名 = build one's fame / rise to fame
如何出名 = how did her build her fame / how did her rise to fame?
a possible answer 1:
She first rose to fame as a singer at the age of 16. 她16岁时便作为歌手一举成名。
In the context, this conversion happens in ancient China, so the language here is slightly more concise than modern Chinese, so '如何出名' here actually means the same as '如何（个）出名（法）', and it means 'in what ways (he or she is) famous or how famous is he(she)', the question is not really specified.
So basically 景容 asked '如何出名' to let 县太爷 describe more details about 他（她）的出名.
And this is like a 口头禅 ————'怎么说'， which is basically ask someone to tell more about what he or she has said.